| L(s) = 1 | + 3·5-s + 7-s + 5·11-s − 6·13-s + 5·17-s + 19-s − 4·23-s + 4·25-s − 6·29-s + 6·31-s + 3·35-s − 8·37-s + 8·41-s + 9·43-s − 47-s − 6·49-s − 2·53-s + 15·55-s + 8·59-s + 11·61-s − 18·65-s + 4·71-s − 11·73-s + 5·77-s − 8·79-s + 4·83-s + 15·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 1.50·11-s − 1.66·13-s + 1.21·17-s + 0.229·19-s − 0.834·23-s + 4/5·25-s − 1.11·29-s + 1.07·31-s + 0.507·35-s − 1.31·37-s + 1.24·41-s + 1.37·43-s − 0.145·47-s − 6/7·49-s − 0.274·53-s + 2.02·55-s + 1.04·59-s + 1.40·61-s − 2.23·65-s + 0.474·71-s − 1.28·73-s + 0.569·77-s − 0.900·79-s + 0.439·83-s + 1.62·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021343894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021343894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.10370248758242841900450466357, −9.753939959817826071873533269689, −9.016095895371464499452424376083, −7.79940297474103721170945704324, −6.90424484022902637686195522952, −5.91683885336442220217491832634, −5.19240240093631142848647003519, −3.99094816844363398828113200246, −2.51416764749634279902449544273, −1.43616825271713767608828534007,
1.43616825271713767608828534007, 2.51416764749634279902449544273, 3.99094816844363398828113200246, 5.19240240093631142848647003519, 5.91683885336442220217491832634, 6.90424484022902637686195522952, 7.79940297474103721170945704324, 9.016095895371464499452424376083, 9.753939959817826071873533269689, 10.10370248758242841900450466357
